Spectral asymptotics for elliptic second order differential operators
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.
scientific article; zbMATH DE number 3964532