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

Abstract

Abstract. We prove the complete asymptotic expansion of the integrated density of states of a two-dimensional Schrödinger operator with a smooth periodic potential.

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

“…However, if the potential b is almost-periodic, we need it to satisfy certain extra conditions; since the formulation of them requires several definitions, we will list these conditions and formulate our main result in the next section. Now we discuss the difference in the approaches of [11] and this paper. To begin with, let us assume that the potential b is periodic.…”

A generalised Gauss circle problem and integrated density of states

“…is unknown), integration over all quasimomenta k ∈ T d := R d /Z d makes things extremely regular, so that there exists a complete asymptotic expansion of N (λ) in powers of λ as λ → ∞, [8,9]. Here, we have denoted (1.10)…”

Sharp asymptotics of the quasimomentum

“…5) The complete asymptotic expansion of the integrated density of states of multidimensional almost-periodic Schrodinger operators were determined by Parnovski, Shterenberg [PS1], see also [KP], [PS2] and the references therein.…”