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,η ) 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 η, KM(H,η )=span η ,Lη ,… ,LM-1η. For a given inner product ( ·, ·) of the operators, the orthonormal basis Oₙ is constructed from O₀=η /√(η ,η ) by Lanczos algorithm. The moments μ ₘ=(O₀,LᵐO₀) are closely related to the important data {bn} called Lanczos coefficients. I present the exact and explicit expressions of the moments {μm} for 16 quantum mechanical systems which are exactly solvable both in the Schrödinger and Heisenberg pictures. The operator η is the variable of the eigenpolynomials. Among them six systems show a clear sign of ‘non-complexity’ as vanishing higher Lanczos coefficients bm = 0, m ≥ 3.