Uniform-in-disorder Wegner bound for the integrated density of states

Establish that the proportionality constant in a Lipschitz upper bound for the integrated density of states of random Schrödinger operators is uniform in the disorder strength in the small-disorder regime; equivalently, prove the conjecture that the integrated density of states satisfies an energy-interval bound whose constant does not diverge as the disorder strength tends to zero.

Background

The paper studies a continuum random Schrödinger operator with a Poisson-distributed random potential in the weak-coupling limit. Standard Wegner estimates yield upper bounds on the integrated density of states that are proportional to the length of the energy interval, but their constants are typically inverse proportional to the disorder strength. Consequently, those estimates become ineffective as the disorder tends to zero.

The cited conjecture asserts that the proportionality constant should remain bounded uniformly in the disorder strength. The paper proves small-disorder asymptotic expansions and derives Hölder continuity estimates with exponents below 2/3, but it does not establish the conjectured uniform Lipschitz bound, so the conjecture remains unresolved in the stated context.