Asymptotic expansion of the spectral function for second-order elliptic operators in \({\mathbb{R}}^ n\)
type: Item
instance of: scholarly article
title: Asymptotic expansion of the spectral function for second-order elliptic operators in \({\mathbb{R}}^ n\)
zbMATH Open document ID: 0533.35072
DOI: 10.1007/BF01078101
publication date: 1983-01-01T00:00:00Z
review text: 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 \(x\neq y\).
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 35J15
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 3846098
zbMATH Keywords: asymptotic expansion
zbMATH Keywords: spectral function
zbMATH Keywords: formally self-adjoint operator
zbMATH Keywords: Laplace-Beltrami operator
author: Georgi Popov
author: M. A. Shubin
MaRDI profile type: Publication
cites work: ON THE SHORT WAVE ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF STATIONARY PROBLEMS AND THE ASYMPTOTIC BEHAVIOUR ASt→ ∞ OF SOLUTIONS OF NON-STATIONARY PROBLEMS
cites work: Q5630230
cites work: An analogue of Weyl's theorem for unbounded domains. I
cites work: Asymptotic behavior of the scattering phase for non-trapping obstacles
cites work: The spectral function of an elliptic operator
cites work: Fourier integral operators. II
cites work: Scattering theory for differential operators. I: Operator theory
cites work: Q5653193
cites work: Curvature and the eigenvalues of the Laplacian
cites work: The spectrum of positive elliptic operators and periodic bicharacteristics
cites work: Q3897569
cites work: Q5644421
cites work: Q3315915
full work available at URL: https://doi.org/10.1007/bf01078101
OpenAlex ID: W2052324451
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 3846098