Localization is well established in the standard Anderson model in the strong
disorder phase. On the other hand, the motivation for the problem, which lies in
many body systems, still lacks a developed theory. We will discuss progress in this
direction, in particular the state labeling method recently developed by Imbrie [I].
As is typical in proof of localization by multiscale analysis, an apriori estimate
to control spectral properties was required, in this case a limited level attraction
estimate. The estimate remains unproven and appeared in [I] as a physically
reasonable assumption. A key difficulty in some many body models such as quantum
spin models is the non monotonicity of spectral energies with respect to random
parameters.
We address this issue in the simplest possible setting, we consider a single body
model with bare energies depending analytically on the random parameters. In
multichannel Schrodinger models, the potentials at each site of the lattice are
matrices which may depend analytically on the random parameters. We will discuss a
method for controlling level attraction which allows a multiscale localization
proof which does not utilize resolvent methods. Our main result is a limited level
attraction estimate [IM] similar to that which appears in [I] as an assumption.

This talk is based on joint work with John Imbrie.

[IM] Imbrie, John Z., and Rajinder Mavi. "Level Spacing for Non-Monotone Anderson
Models." arXiv preprint arXiv:1506.06692 (2015).

[I] Imbrie, John Z. "On many-body localization for quantum spin chains." arXiv
preprint arXiv:1403.7837 (2014).