Integrated Density of States for the Periodic Schrödinger Operator in Dimension Two

Abstract

Here H D is the restriction of H to the cube [0, R] d with the Dirichlet boundary conditions, and N(λ; · ) is the counting function of the discrete spectrum of H D . The above limit exists for periodic and almost periodic potentials, see [17], [22]. To be precise, the quantity D(λ) is called the integrated density of states, but for the sake of brevity we call it simply the density of states. Calculation of the density of states D0(λ) for the unperturbed operator H0 = −∆ is an elementary exercise: one easily proves (see, e.g., Proposition 2.4 below) that