Asymptotic behaviour of the spectral function of an elliptic equat

In this paper we consider the asymptotic behaviour of the spectral function of an elliptic differential (pseudodifferential) equation or system of equations.
For the case of differential operators this problem has been widely studied, and various methods have been developed for its solution (see [1] and [2] for a urvey of these methods).
We consider in this paper just one of these methods. It is based on a study of the structure of the fundamental solution of the Cauchy problem for a hyperbolic differential (pseudodifferential) equation. The method we use is called “the method of geometrical optics”. For a system of first order differential equations it was originally developed in detail by Lax [8], and for pseudodifferential equations by Hörmander [2] and independently by Eskin [17], [18] and Maslov [19].
In [2] Hörmander also investigates the asymptotic behaviour of the spectral function for an elliptic pseudodifferential first order operator. Using some important results of Seeley [5] one can then derive the asymptotic behaviour of the spectral function of an elliptic differential operator of arbitrary order.
Similar methods have previously been applied by the author in [3], [4] for second order elliptic differential operators.
In this article we give a partial account of the results of [8] and [9]. We also present some new results due to the author, concerning both the structure of the fundamental solution of the Cauchy problem and the asymptotic behaviour of the spectral function.