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In this article, we consider the asymptotic behaviour of the spectralfunction 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 withsmooth coefficients, bounded with all derivatives. We show that the kernel ofthe spectral projector, $\mathbb{1}_{(-\infty,\rho^2]}(H)$, has a completeasymptotic expansion in powers of $\rho$. This settles the 1-dimensional caseof a conjecture made by the last two authors.

Classical Wave methods and modern gauge transforms: Spectral Asymptotics in the one dimensional case