A complete asymptotic expansion of a spectral function of elliptic operators in \(R^ n\)
Let L be a second order selfadjoint elliptic differential operator in \(R^ n\) with smooth coefficients and \(L=-\Delta\) for \(| x|\gg 1\). Let \(\{E_{\lambda}\}\) be the spectral family of the operator L and e'\({}_{\lambda}=e'\!_{\lambda}(x,y)\) a kernel of the operator \(dE_{\lambda}/d\lambda\). A complete asymptotic expansion of the function e'\({}_{\lambda}\) at \(\lambda \to\infty \), \(| x|,| y|\leq b\) with arbitrary b is given. The function e'\({}_{\lambda}\) is asymptotically presented as an integral over \(S^{n-1}\) of a certain family of Maslov's canonical operators, which depend on a parameter \(w\in S^{n-1}\). The only condition on L is that of non-trapping, which means that all the rays of geometrical optics escape to infinity (the family of rays starting from a point y may have caustics and geodesic loops).
second order selfadjoint elliptic differential operator
scientific article; zbMATH DE number 3871880