Towards verifications of Krylov complexity

Abstract

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 $b_m=0$, $m\ge3$.