A complete asymptotic expansion of a spectral function of elliptic operators in \(R^ n\)
type: Item
instance of: scholarly article
title: A complete asymptotic expansion of a spectral function of elliptic operators in \(R^ n\)
zbMATH Open document ID: 0547.35040
published in: Moscow University Mathematics Bulletin
publication date: 1983-01-01T00:00:00Z
review text: 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).
Mathematics Subject Classification ID: 35J15
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 35C20
zbMATH DE Number: 3871880
zbMATH Keywords: second order selfadjoint elliptic differential operator
zbMATH Keywords: smooth coefficients
zbMATH Keywords: complete asymptotic expansion
zbMATH Keywords: Maslov's canonical operators
author: B. Vainberg
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3871880