Bounds on the density of states for Schr\" odinger operators

We establish bounds on the density of states measure for Schr\"odinger operators. These are deterministic results that do not require the existence of the density of states measure, or, equivalently, of the integrated density of states. The results are stated in terms of a "density of states outer-measure" that always exists, and provides an upper bound for the density of states measure when it exists. We prove log-H\"older continuity for this density of states outer-measure in one, two, and three dimensions for Schr\"odinger operators, and in any dimension for discrete Schr\"odinger operators.