Prethermalization for Deformed Wigner Matrices

We prove that a class of weakly perturbed Hamiltonians of the form $H_{\lambda }=H_0+\lambda W$, with $W$ being a Wigner matrix, exhibits prethermalization. That is, the time evolution generated by $H_{\lambda }$ relaxes to its ultimate thermal state via an intermediate prethermal state with a lifetime of order ${\lambda }^{-2}$. Moreover, we obtain a general relaxation formula, expressing the perturbed dynamics via the unperturbed dynamics and the ultimate thermal state. The proof relies on a two-resolvent law for the deformed Wigner matrix $H_{\lambda }$.