Complete asymptotic expansions of the spectral function for symbolic perturbations of almost periodic Schrödinger operators in dimension one

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