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}(mathbb{R})to L^{2}(mathbb{R}) have the form \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ H:=-\\frac{d^{2}}{dx^{2}}+Q, $$\\end{document} 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.