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.

中文翻译：

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经典波方法和现代规范变换：一维情况下的谱渐近

在本文中，我们考虑薛定谔算子谱函数在实线上的渐近行为。设\(H: L^{2}(\mathbb{R})\to L^{2}(\mathbb{R})\)具有以下形式

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

其中Q是具有平滑系数的形式自伴一阶微分算子，以所有导数为界。我们证明光谱投影仪的内核\({1}_{(-\infty ,\rho ^{2}]}(H)\)具有ρ幂的完全渐近展开式。这解决了 1最后两位作者提出的猜想的维案例。