Simon's Problem 2: pure-point spectrum for the two-dimensional Anderson model at every disorder

For each fixed $h>0$, the operator $H_v=\Delta+v$ on $\ell^2(\mathbb{Z}^2)$ with iid $v_x$ uniform on $[-h,h]$ almost surely has pure-point spectral type with a complete orthonormal eigenbasis, spectrum $[-4-h,4+h]$ and dense eigenvalues. The null set may depend on $h$. Not shown: one event for all $h$, bounds uniform in energy, exponential decay of eigenfunctions, dynamical localization, or other single-site laws such as Bernoulli.

For each fixed $h>0$, the operator $H_v=\Delta+v$ on $\ell^2(\mathbb{Z}^2)$ with iid $v_x$ uniform on $[-h,h]$ almost surely has pure-point spectral type with a complete orthonormal eigenbasis, spectrum $[-4-h,4+h]$ and dense eigenvalues. The null set may depend on $h$. Not shown: one event for all $h$, bounds uniform in energy, exponential decay of eigenfunctions, dynamical localization, or other single-site laws such as Bernoulli.