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

Complete asymptotic expansion of the integrated density of states of multidimensional almost-periodic Schrödinger operators

“…Unfortunately, this approach does not work if d ≥ 3, since then the cluster multiplicity of P BP becomes unbounded. The second method of obtaining asymptotic formulas for the eigenvalues of H(k) was developed in [19] and [20] and was also used in [12]. This method, which we call the gauge transform method, consists of constructing two pseudodifferential operators, H 1 and H 2 .…”

“…In Section 5, we define resonance regions and prove their properties. The reader who has read several of the papers [20], [10], [1], [11], [12] may have noticed that in each of these papers the construction of the resonance regions is slightly different. The reason is that each time we define these regions, we need to fine tune the definition taking care of the problem we are trying to solve.…”

Asymptotic expansion of the integrated density of states of a two-dimensional periodic Schrödinger operator

“…In the multidimensional case, only partial results are known, see [1], [4], [5], [9], [12], [13]. In particular, in [13] it was shown that when d = 2 formula (1.6) is valid with K = 2 and R(λ) = O(λ − 6 5 +ǫ ) for any positive ǫ; in [4] it was shown that when d ≥ 3 formula (1.6) is valid with K = 1 and R(λ) = O(λ −δ ) with some small δ when d = 3 and R(λ) = O(λ d−3 2 ln λ) when d > 3. The aim of this paper is to establish the complete asymptotic formula (1.5) in the 2-dimensional case.…”

Complete Asymptotic Expansion of the Integrated Density of States of Multidimensional Almost-Periodic Pseudo-Differential Operators

“…The approach of our paper is similar to the one of [5]. In particular, we use the method of gauge transform developed in [9], [10], and [6]. Nevertheless, there are plenty of new (mostly technical, but sometimes ideological) difficulties arising because the operator B is no longer bounded and no longer local.…”