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

$$ H:=-\frac{d^{2}}{dx^{2}}+Q, $$

where Q 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, \({1}_{(-\infty ,\rho ^{2}]}(H)\), has a complete asymptotic expansion in powers of ρ. This settles the 1-dimensional case of a conjecture made by the last two authors.