We show that there is a countable dense set of energies at which the integrated density of states of the 1D discrete Anderson–Bernoulli model can be given explicitly and does not depend on the disorder parameter, provided that the latter is above an energy-dependent threshold.

In this article, we give upper and lower bounds for the integrated density of states (IDS) of the 1D discrete Anderson–Bernoulli model when the disorder is strong enough to separate the two spectral bands. These bounds are uniform on the disorder and hold over the whole spectrum. They show the existence of a sequence of energies in which the value of the IDS can be given explicitly and does not depend on the disorder parameter.

We state a precise formulation of a conjecture concerning the product of the principal eigenvalue and the sup-norm of the landscape function of the discrete Anderson model restricted to a large box. We first provide the asymptotic of the principal eigenvalue as the size of the box grows, and then use it to give a partial proof of the conjecture. For the one dimensional case, we give a complete proof by means of Green function bounds.