Complete Semiclassical Spectral Asymptotics for Periodic and Almost Periodic Perturbations of Constant Operators

TL;DR

This paper derives comprehensive semiclassical spectral asymptotics for scalar operators with periodic or almost periodic perturbations, advancing understanding of spectral functions and density of states in such settings.

Contribution

It provides a complete semiclassical asymptotic expansion for the spectral function and integrated density of states for operators with periodic or almost periodic perturbations.

Findings

Derived full semiclassical asymptotics of spectral function

Established asymptotics for integrated density of states

Extended results to more general cases

Abstract

Under certain assumptions we derive a complete semiclassical asymptotics of the spectral function $e_{h,\varepsilon}(x,x,\lambda)$ for a scalar operator \begin{equation*} A_\varepsilon (x,hD)= A^0(hD) + \varepsilon B(x,hD), \end{equation*} where $A^0$ is an elliptic operator and $B(x,hD)$ is a periodic or almost periodic perturbation. In particular, a complete semiclassical asymptotics of the integrated density of states also holds. Further, we consider generalizations.