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 η, $\mathcal {K}_M(\mathcal {H},\eta )=\text{span}\lbrace \eta ,\mathcal {L}\eta ,\ldots ,\mathcal {L}^{M-1}\eta \rbrace$. For a given inner product ( ·, ·) of the operators, the orthonormal basis $\lbrace \mathcal {O}_n\rbrace$ 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 {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.