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

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