We show absence of energy levels repulsion for the eigenvalues of random Schrödinger operators in the continuum. We prove that, in the localization region at the bottom of the spectrum, the properly rescaled eigenvalues of a continuum Anderson Hamiltonian are distributed as a Poisson point process with intensity measure given by the density of states. In addition, we prove that in this localization region the eigenvalues are simple.
These results rely on a Minami estimate for continuum Anderson Hamiltonians. We also give a simple, transparent proof of Minami’s estimate for the (discrete) Anderson model.
"Poisson statistics for eigenvalues of continuum random Schrödinger operators." Anal. PDE 3 (1) 49 - 80, 2010. https://doi.org/10.2140/apde.2010.3.49