In this article we consider asymptotics for the spectral function of Schrödinger operators on the real line. Let P\colon L^2(\mathbb{R})\to L^2(\mathbb{R}) have the 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 the kernel of the spectral projector, \mathbf{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, the class of potentials 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.