Complete asymptotic expansions of the spectral function for symbolic perturbations of almost periodic Schr\"odinger operators in dimension one

TL;DR

This paper derives complete asymptotic expansions for the spectral function of one-dimensional Schr"odinger operators with almost periodic perturbations, including stability under certain perturbations and potentials with dense spectra.

Contribution

It provides the first full asymptotic expansion of the spectral function for a broad class of almost periodic Schr"odinger operators in one dimension, including stability results.

Findings

Full asymptotic expansion in powers of or spectral projector kernel.

Stability of the class of potentials under certain perturbations.

Includes potentials with dense pure point spectrum.

Abstract

In this article we consider asymptotics for the spectral function of Schr\"odinger operators on the real line. Let $P:L^2(\mathbb{R})\to L^2(\mathbb{R})$ have the form $$ P:=-\tfrac{d^2}{dx^2}+W, $$ where $W$ is a self-adjoint first order differential operator with certain modified almost periodic structure. We show that the kernel of the spectral projector, $\mathbb{1}_{(-\infty,\lambda^2]}(P)$ has a full asymptotic expansion in powers of $\lambda$. In particular, our class of potentials $W$ is stable under perturbation by formally self-adjoint first order differential operators with smooth, compactly supported coefficients. Moreover, it includes certain potentials with dense pure point spectrum. The proof combines the gauge transform methods of Parnovski-Shterenberg and Sobolev with Melrose's scattering calculus.