Thermalization in Krylov Basis

We study thermalization for closed non-integrable quantum systems in the Krylov basis. Following the idea of the eigenstate thermalization hypothesis, one may introduce Krylov basis thermalization hypothesis which imposes a condition on the matrix elements of local operators in the Krylov basis. In this context, the nature of thermalization may be probed by the infinite time average of the Krylov complexity. We also compute the variance of Lanczos coefficients which may provide another quantity to examine the nature of thermalization. We will see that there is a direct relation between the behavior of the infinite time average of complexity and that of the inverse participation ratio of initial states.