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

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}$)$havetheform$$P:=-\frac{d^2}{d{x}^2}+W,$$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, ${\mathbb{1}}_{(-\mathrm{\infty },{\lambda }^2]}(P)$ 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.