Spectral asymptotics for elliptic second order differential operators
type: Item
instance of: scholarly article
title: Spectral asymptotics for elliptic second order differential operators
zbMATH Open document ID: 0598.35081
published in: Journal of Mathematics of Kyoto University
publication date: 1985-01-01T00:00:00Z
review text: Let X be an open set in \(R^ n\), \(n\geq 2\), with a smooth boundary and let \(R^ n-X\) be bounded. Let \(R^ n\) be equipped with a Riemann metric \(g_{ij}\), which is Euclidean in a neighborhood of infinity and let H be a self-adjoint operator in \(L^ 2(X)\); \(H=-\Delta_ g+V(x)\) with Dirichlet or Neumann boundary conditions, where \(\Delta_ g\) is the Laplace-Beltrami operator and V is a smooth function with a compact support. Under some additional assumptions on the geometry of X the author proves the asymptotic expansions of \(G^{\pm}(\lambda;x,y)\) and \((d/d\lambda)e(\lambda^ 2;x,y)\), as \(\lambda\) \(\to \infty\), where e(\(\lambda\) ;x,y) is the spectral function of H and \(G^{\pm}(\lambda;x,y)\) are the Green's functions of \((H-\lambda^ 2)\) satisfying suitable Sommerfeld's conditions at infinity. The proof is based on an integral representation of Green's functions.
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 35J15
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 3964532
zbMATH Keywords: fundamental solution
zbMATH Keywords: Fourier distribution
zbMATH Keywords: Riemann metric
zbMATH Keywords: Laplace- Beltrami operator
zbMATH Keywords: spectral function
zbMATH Keywords: Green's functions
zbMATH Keywords: Sommerfeld's conditions
author: Georgi Popov
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250521014
OpenAlex ID: W1566493562
DOI: 10.1215/KJM/1250521014
description: scientific article; zbMATH DE number 3964532