Asymptotic expansion of the spectral function for second-order elliptic operators in \({\mathbb{R}}^ n\)
Let \(H=-\Delta_ g+i<h,\nabla>+V(x)\), \(h,V\in C_ 0\!^{\infty}(R^ n)\) be a formally self-adjoint operator. Here \(\Delta_ g\) is the Laplace-Beltrami operator related to a Riemannian metric \(g_{ij}\) which is Euclidean near the infinity. In this paper the asymptotics of the spectral function \(e(\lambda\),x,y) is investigated as \(\lambda\to \infty\). The geodesics of the metric \(g_{ij}\) are supposed to go to infinity and to have no self-intersections. As a consequence a complete asymptotic expansion of \(e(\lambda\),x,y) as \(\lambda\to \infty\) is obtained in both cases \(x=y\) and \(x\neq y\).
scientific article; zbMATH DE number 3846098
scientific article