We prove that a class of weakly perturbed Hamiltonians of the form Hλ=H0+λW, with W being a Wigner matrix, exhibits prethermalization.
That is, the time evolution generated by Hλ relaxes to its ultimate
thermal state via an intermediate prethermal state with a lifetime of order
λ−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
Hλ.Comment: 31 pages (including appendix), 3 figure