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

TL;DR

This paper analyzes the spectral asymptotics of one-dimensional Schrödinger operators with smooth, bounded potentials, providing a complete asymptotic expansion of the spectral projector kernel, confirming a conjecture in the field.

Contribution

It establishes a full asymptotic expansion for the spectral projector kernel of 1D Schrödinger operators, advancing understanding of spectral asymptotics in this setting.

Findings

Complete asymptotic expansion of the spectral projector kernel in powers of 

Confirmation of a conjecture regarding spectral asymptotics in 1D

Extension of classical wave methods to modern gauge transforms

Abstract

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