  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
$Q=(1+|x|^2)(1-\Delta)$ on $\mathbb{R}^d$.
