Problem of focusing and the asymptotics of the spectral function of the Laplace-Beltrami operator. I

Abstract

The connection between the asymptotics of the spectral function and the formal shortwave expansion of the solution of the problem of the asymptotics of the Green function near a geodesically concave boundary of a two-dimensional surface is considered in the paper.By means of theorems of Tauberian type the asymptotic behavior of the spectral function of elliptic differential operators reduces to the study of stationary and nonstationary Green functions of wave problems, i.e., to questions with which the mathematical theo1~ of diffraction is concerned.In the theory of diffraction there are many results and techniques which are little known among specialists in the spectral theory of differential operators.Use of the methods of diffraction theory makes it possible, in particular, under certain additional conditions to extend the classical asymptotic formulas for the spectral function of the Laplace operator to a neighborhood of the boundary. This is the topic of our paper. A brief exposition of our results is contained in [i].