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(ℝ)→ L^2(ℝ) have the form H:=-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_(-∞ ,ρ ^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.