  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 $\mathcal{K}_M(\mathcal{H},\eta)$
spanned by the multiple applications of the Liouville operator $\mathcal{L}$
defined by the commutator in terms of a Hamiltonian $\mathcal{H}$,
$\mathcal{L}:=[\mathcal{H},\cdot]$ acting on an operator $\eta$,
$\mathcal{K}_M(\mathcal{H},\eta)=\text{span}\{\eta,\mathcal{L}\eta,\ldots,\mathcal{L}^{M-1}\eta\}$.
For a given inner product $(\cdot,\cdot)$ of the operators, the orthonormal
basis $\{\mathcal{O}_n\}$ is constructed from
$\mathcal{O}_0=\eta/\sqrt{(\eta,\eta)}$ by Lanczos algorithm. The moments
$\mu_m=(\mathcal{O}_0,\mathcal{L}^m\mathcal{O}_0)$ are closely related to the
important data $\{b_n\}$ called Lanczos coefficients. I present the exact and
explicit expressions of the moments $\{\mu_m\}$ for 16 quantum mechanical
systems which are {\em exactly solvable both in the Schr\"odinger and
Heisenberg pictures}. The operator $\eta$ is the variable of the
eigenpolynomials. Among them six systems show a clear sign of `non-complexity'
as vanishing higher Lanczos coefficients $b_m=0$, $m\ge3$.
