Spectral shift function for slowly varying perturbation of periodic Schroedinger operators

TL;DR

This paper investigates the asymptotic behavior of the spectral shift function for slowly varying perturbations of periodic Schrödinger operators, providing detailed expansions and establishing a limiting absorption theorem.

Contribution

It introduces new asymptotic expansions for the spectral shift function derivative and proves a limiting absorption theorem for these perturbed operators.

Findings

Derived weak and pointwise asymptotics in powers of h

Established a limiting absorption theorem for P(h)

Provided detailed asymptotic expansions of the spectral shift function

Abstract

In this paper we study the asymptotic expansion of the spectral shift function for the slowly varying perturbations of periodic Schr\"odinger operators. We give a weak and pointwise asymptotics expansions in powers of $h$ of the derivative of the spectral shift function corresponding to the pair $\big(P(h)=P_0+\phi(hx),P_0=-\Delta+V(x)\big),$ where $\phi(x)\in {\mathcal C}^\infty(\mathbb R^n,\mathbb R)$ is a decreasing function, ${\mathcal O}(|x|^{-\delta})$ for some $\delta>n$ and $h$ is a small positive parameter. Here the potential $V$ is real, smooth and periodic with respect to a lattice $\Gamma$ in ${\mathbb R}^n$. To prove the pointwise asymptotic expansion of the spectral shift function, we establish a limiting absorption Theorem for $P(h)$.