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

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