This article is cited in 6 scientific papers (total in 6 papers)
Asymptotic behaviour of the spectral function of an elliptic equat
B. M. Levitan
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  and  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 , and for pseudodifferential equations by Hörmander  and independently by
Eskin ,  and Maslov .
In  Hörmander also investigates the asymptotic behaviour of the spectral function for an elliptic pseudodifferential first order operator. Using some important results of Seeley  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 ,  for second order elliptic differential operators.
In this article we give a partial account of the results of  and . 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.
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Russian Mathematical Surveys, 1971, 26:6, 165–232
MSC: 35B40, 35J45, 35J15, 35P05, 35Sxx, 35A08
B. M. Levitan, “Asymptotic behaviour of the spectral function of an elliptic equat”, Uspekhi Mat. Nauk, 26:6(162) (1971), 151–212; Russian Math. Surveys, 26:6 (1971), 165–232
Citation in format AMSBIB
\paper Asymptotic behaviour of the spectral function of an elliptic equat
\jour Uspekhi Mat. Nauk
\jour Russian Math. Surveys
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