  In this article, we consider the asymptotic behaviour of the spectral
function of Schr\"odinger operators on the real line. Let $H:
L^2(\mathbb{R})\to L^2(\mathbb{R})$ have the form $$ H:=-\frac{d^2}{dx^2}+V, $$
where $V$ is a formally self-adjoint first order differential operator with
smooth coefficients, bounded with all derivatives. We show that the kernel of
the spectral projector, $\mathbb{1}_{(-\infty,\rho^2]}(H)$, has a complete
asymptotic expansion in powers of $\rho$. This settles the 1-dimensional case
of a conjecture made by the last two authors.
