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

Abstract

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 formwhere 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, 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.