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 ${\mathcal{K}}_M(\mathcal{H},\hat{I}\cdot )$ 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 $\hat{I}\cdot$, ${\mathcal{K}}_M(\mathcal{H},\hat{I}\cdot )=\text{span}\{\hat{I}\cdot ,\mathcal{L}\hat{I}\cdot ,\dots ,{\mathcal{L}}^{M-1}\hat{I}\cdot \}$. For a given inner product $(\cdot ,\cdot )$ of the operators, the orthonormal basis $\{{\mathcal{O}}_n\}$ is constructed from ${\mathcal{O}}_0=\hat{I}\cdot /\sqrt{(\hat{I}\cdot ,\hat{I}\cdot )}$ by Lanczos algorithm. The moments $\hat{I}{¼}_m=({\mathcal{O}}_0,{\mathcal{L}}^m{\mathcal{O}}_0)$ are closely related to the important data $\{b_n\}$ called Lanczos coefficients. I present the exact and explicit expressions of the moments $\{\hat{I}{¼}_m\}$ for 16 quantum mechanical systems which are {\em exactly solvable both in the Schrödinger and Heisenberg pictures}. The operator $\hat{I}\cdot$ is the variable of the eigenpolynomials. Among them six systems show a clear sign of `non-complexity' as vanishing higher Lanczos coefficients $b_m=0$, $m\ge 3$.

typos and errors are corrected, LaTeX 29pages, no figure