Finite-time quantum equilibration for continuous variables

Leveraging the techniques found in the literature on Quantum Equilibration for finite dimensional systems, we develop the theory of Quantum Equilibration for the case of infinite-dimensional systems, particularly the cases where the dynamics-generating Hamiltonians have continuous spectrum. The main goal of this paper will be to propose a framework to extend the results obtained by Short in, where estimates for the equilibration-on-average and effective equilibration for the case of Hamiltonians with continuous spectrum are derived. We will show that in the latter setting, it is compulsory to constrain ourselves to finite time equilibration; we then develop estimates analogous to the main results in the proposed setting.