We study the asymptotic behaviour of the eigenvalue counting function for self-adjoint elliptic linear operators defined through classical weighted symbols of order (1, 1), on an asymptotically Euclidean manifold. We first prove a two-term Weyl formula, improving previously known remainder estimates. Subsequently, we show that under a geometric assumption on the Hamiltonian flow at infinity, there is a refined Weyl asymptotics with three terms. The proof of the theorem uses a careful analysis of the flow behaviour in the corner component of the boundary of the double compactification of the cotangent bundle. Finally, we illustrate the results by analysing the operator \(Q=(1+|x|^2)(1-\varDelta )\) on \(\mathbb {R}^d\).

我们研究了通过渐近欧几里得流形上通过阶次（1，1）的经典加权符号定义的自伴随椭圆线性算子的特征值计数函数的渐近行为。我们首先证明了两项Weyl公式，从而改进了先前已知的余数估计。随后，我们证明了在无穷阶哈密顿流的几何假设下，存在带有三个项的精炼的魏尔渐近线。该定理的证明使用了对切向束的双重压实边界的角部流动特性的仔细分析。最后，我们通过分析\（\ mathbb {R} ^ d \）上的运算符\（Q =（1+ | x | ^ 2）（1- \ varDelta）\）来说明结果。