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.
Keywords
Anderson localization, Poisson statistics of eigenvalues,
Minami estimate, level statistics