Jak\v{s}i\'{c}-Last Theorem for Higher Rank Perturbations

We consider the generalized Anderson Model $\Delta+\sum_{n\in\mathcal{N}}\omega_n P_n$, where $\mathcal{N}$ is a countable set, $\{\omega_n\}_{n\in\mathcal{N}}$ are i.i.d random variables and $P_n$ are rank $N<\infty$ projections. For these models we prove theorem analogous to that of Jak\v{s}i\'{c}-Last on the equivalence of the trace measure $\sigma_n(\cdot)=tr(P_nE_{H^\omega}(\cdot)P_n)$ for $n\in\mathcal{N}$ a.e $\omega$. Our model covers the dimer and polymer models.