Loschmidt Echo for Deformed Wigner Matrices

We consider two Hamiltonians that are close to each other, $H_1\approx H_2$, and analyze the time-decay of the corresponding Loschmidt echo $\mathfrak{M}(t):=\mathrm{\mid }⟨{\psi }_0,{\mathrm{e}}^{\mathrm{i}t{H}_2}{\mathrm{e}}^{-\mathrm{i}t{H}_1}{\psi }_0⟩{\mathrm{\mid }}^2$ that expresses the effect of an imperfect time reversal on the initial state ${\psi }_0$. Our model Hamiltonians are deformed Wigner matrices that do not share a common eigenbasis. The main tools for our results are two-resolvent laws for such $H_1$ and $H_2$.