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

In this article, we consider the asymptotic behaviour of the spectral function of Schrödinger operators on the real line. Let $H:L^2(\mathbb{R})\to L^2(\mathbb{R})$ have the form $$H:=-\frac{d^2}{d{x}^2}+V,$$ 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, ${\mathbb{1}}_{(-\mathrm{\infty },{\rho }^2]}(H)$, has a complete asymptotic expansion in powers of $\rho$. This settles the 1-dimensional case of a conjecture made by the last two authors.