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 \(H_\lambda \) .
